Note on $ r $-central Lah numbers and $ r $-central Lah-Bell numbers

Hye Kyung Kim; Hye Kyung Kim

doi:10.3934/math.2022161

AIMS Mathematics

2022, Volume 7, Issue 2: 2929-2939. doi: 10.3934/math.2022161

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Research article Special Issues

Note on $ r $-central Lah numbers and $ r $-central Lah-Bell numbers

Hye Kyung Kim ^,

Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea

Received: 21 May 2021 Accepted: 16 November 2021 Published: 22 November 2021
MSC : 11F20, 11B68, 11B83

The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.
- Lah numbers,
- Lah-Bell numbers,
- $ r $-Lah numbers,
- $ r $-Lah-Bell polynomials,
- central factorial numbers of the second kind
Citation: Hye Kyung Kim. Note on $ r $-central Lah numbers and $ r $-central Lah-Bell numbers[J]. AIMS Mathematics, 2022, 7(2): 2929-2939. doi: 10.3934/math.2022161

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Abstract

The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.

References

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